18,519 research outputs found
Asymptotic Estimates in Information Theory with Non-Vanishing Error Probabilities
This monograph presents a unified treatment of single- and multi-user
problems in Shannon's information theory where we depart from the requirement
that the error probability decays asymptotically in the blocklength. Instead,
the error probabilities for various problems are bounded above by a
non-vanishing constant and the spotlight is shone on achievable coding rates as
functions of the growing blocklengths. This represents the study of asymptotic
estimates with non-vanishing error probabilities.
In Part I, after reviewing the fundamentals of information theory, we discuss
Strassen's seminal result for binary hypothesis testing where the type-I error
probability is non-vanishing and the rate of decay of the type-II error
probability with growing number of independent observations is characterized.
In Part II, we use this basic hypothesis testing result to develop second- and
sometimes, even third-order asymptotic expansions for point-to-point
communication. Finally in Part III, we consider network information theory
problems for which the second-order asymptotics are known. These problems
include some classes of channels with random state, the multiple-encoder
distributed lossless source coding (Slepian-Wolf) problem and special cases of
the Gaussian interference and multiple-access channels. Finally, we discuss
avenues for further research.Comment: Further comments welcom
A Formula for the Capacity of the General Gel'fand-Pinsker Channel
We consider the Gel'fand-Pinsker problem in which the channel and state are
general, i.e., possibly non-stationary, non-memoryless and non-ergodic. Using
the information spectrum method and a non-trivial modification of the piggyback
coding lemma by Wyner, we prove that the capacity can be expressed as an
optimization over the difference of a spectral inf- and a spectral sup-mutual
information rate. We consider various specializations including the case where
the channel and state are memoryless but not necessarily stationary.Comment: Accepted to the IEEE Transactions on Communication
Second-Order Asymptotics for the Classical Capacity of Image-Additive Quantum Channels
We study non-asymptotic fundamental limits for transmitting classical
information over memoryless quantum channels, i.e. we investigate the amount of
classical information that can be transmitted when a quantum channel is used a
finite number of times and a fixed, non-vanishing average error is permissible.
We consider the classical capacity of quantum channels that are image-additive,
including all classical to quantum channels, as well as the product state
capacity of arbitrary quantum channels. In both cases we show that the
non-asymptotic fundamental limit admits a second-order approximation that
illustrates the speed at which the rate of optimal codes converges to the
Holevo capacity as the blocklength tends to infinity. The behavior is governed
by a new channel parameter, called channel dispersion, for which we provide a
geometrical interpretation.Comment: v2: main results significantly generalized and improved; v3: extended
to image-additive channels, change of title, journal versio
Asymmetric Evaluations of Erasure and Undetected Error Probabilities
The problem of channel coding with the erasure option is revisited for
discrete memoryless channels. The interplay between the code rate, the
undetected and total error probabilities is characterized. Using the
information spectrum method, a sequence of codes of increasing blocklengths
is designed to illustrate this tradeoff. Furthermore, for additive discrete
memoryless channels with uniform input distribution, we establish that our
analysis is tight with respect to the ensemble average. This is done by
analysing the ensemble performance in terms of a tradeoff between the code
rate, the undetected and the total errors. This tradeoff is parametrized by the
threshold in a generalized likelihood ratio test. Two asymptotic regimes are
studied. First, the code rate tends to the capacity of the channel at a rate
slower than corresponding to the moderate deviations regime. In this
case, both error probabilities decay subexponentially and asymmetrically. The
precise decay rates are characterized. Second, the code rate tends to capacity
at a rate of . In this case, the total error probability is
asymptotically a positive constant while the undetected error probability
decays as for some . The proof techniques involve
applications of a modified (or "shifted") version of the G\"artner-Ellis
theorem and the type class enumerator method to characterize the asymptotic
behavior of a sequence of cumulant generating functions.Comment: 28 pages, no figures in IEEE Transactions on Information Theory, 201
Second-Order Asymptotics for the Discrete Memoryless MAC with Degraded Message Sets
This paper studies the second-order asymptotics of the discrete memoryless
multiple-access channel with degraded message sets. For a fixed average error
probability and an arbitrary point on the boundary of the
capacity region, we characterize the speed of convergence of rate pairs that
converge to that point for codes that have asymptotic error probability no
larger than , thus complementing an analogous result given previously
for the Gaussian setting.Comment: 5 Pages, 1 Figure. Follow-up paper of http://arxiv.org/abs/1310.1197.
Accepted to ISIT 201
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